Python.h missing in Blender Python?
First, if you install blender with your linux distro,
then a proper installation of python3 and blender with your particular distro should solve your problem.   

e.g. for debian:


sudo apt-get install python3 python3-dev 
sudo apt-get install blender


Second, if you really want to install the python header files
for the python that runs inside blender, then you can do it the following way:

Find out the exact blender python version:


/path/to/blender-2.78a-linux-glibc211-x86_64/2.78/python/bin/python3.5m -V


Download and install the python header files from the official python homepage:


point your browser at https://www.python.org/downloads/source/
download Gzipped source tarball from the exact same python version as your blender python version
cd ~/Downloads/
tar -xzf Python-3.X.X.tgz
cp Python-3.X.X/Include/* /path/to/blender-2.78a-linux-glibc211-x86_64/2.78/python/include/python3.5m/


That should get you rolling. 
Best, Bue